(* Title: HOL/TLA/Init.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
Introduces type of temporal formulas. Defines interface between
temporal formulas and its "subformulas" (state predicates and
actions).
*)
theory Init
imports Action
begin
typedecl behavior
arities behavior :: world
type_synonym temporal = "behavior form"
consts
Initial :: "('w::world => bool) => temporal"
first_world :: "behavior => ('w::world)"
st1 :: "behavior => state"
st2 :: "behavior => state"
syntax
"_TEMP" :: "lift => 'a" ("(TEMP _)")
"_Init" :: "lift => lift" ("(Init _)"[40] 50)
translations
"TEMP F" => "(F::behavior => _)"
"_Init" == "CONST Initial"
"sigma |= Init F" <= "_Init F sigma"
defs
Init_def: "sigma |= Init F == (first_world sigma) |= F"
fw_temp_def: "first_world == %sigma. sigma"
fw_stp_def: "first_world == st1"
fw_act_def: "first_world == %sigma. (st1 sigma, st2 sigma)"
lemma const_simps [int_rewrite, simp]:
"|- (Init #True) = #True"
"|- (Init #False) = #False"
by (auto simp: Init_def)
lemma Init_simps1 [int_rewrite]:
"!!F. |- (Init ~F) = (~ Init F)"
"|- (Init (P --> Q)) = (Init P --> Init Q)"
"|- (Init (P & Q)) = (Init P & Init Q)"
"|- (Init (P | Q)) = (Init P | Init Q)"
"|- (Init (P = Q)) = ((Init P) = (Init Q))"
"|- (Init (!x. F x)) = (!x. (Init F x))"
"|- (Init (? x. F x)) = (? x. (Init F x))"
"|- (Init (?! x. F x)) = (?! x. (Init F x))"
by (auto simp: Init_def)
lemma Init_stp_act: "|- (Init $P) = (Init P)"
by (auto simp add: Init_def fw_act_def fw_stp_def)
lemmas Init_simps2 = Init_stp_act [int_rewrite] Init_simps1
lemmas Init_stp_act_rev = Init_stp_act [int_rewrite, symmetric]
lemma Init_temp: "|- (Init F) = F"
by (auto simp add: Init_def fw_temp_def)
lemmas Init_simps = Init_temp [int_rewrite] Init_simps2
(* Trivial instances of the definitions that avoid introducing lambda expressions. *)
lemma Init_stp: "(sigma |= Init P) = P (st1 sigma)"
by (simp add: Init_def fw_stp_def)
lemma Init_act: "(sigma |= Init A) = A (st1 sigma, st2 sigma)"
by (simp add: Init_def fw_act_def)
lemmas Init_defs = Init_stp Init_act Init_temp [int_use]
end